Vector Linear Solution iff Dimension \geq m

• Published in: IEEE communications letters Vol. 23; no. 9; pp. 1470 - 1473
• Main Authors: Das, Niladri, Rai, Brijesh Kumar
• Format: Journal Article
• Language: English
• Published: IEEE 01.09.2019
• Subjects: Indexes; Knowledge engineering; Linear network coding; M-network; message dimension; Network coding; Routing; scalar linear solution; Silicon; Sun; vector linear solution; Zinc
• ISSN: 1089-7798; 1558-2558
• DOI: 10.1109/LCOMM.2019.2922655

If a network has an <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>-dimensional vector linear network coding (VLNC) solution, then it also has a VLNC solution for all dimensions multiple of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>; but can a VLNC solution exists for dimensions not a multiple of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>? If <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> is the least positive integer such that a network has an <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>-dimensional VLNC solution, none of the networks shown in the literature has a VLNC solution for a dimension which is not a multiple of <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>. In this letter, we show that for any positive integer <inline-formula> <tex-math notation="LaTeX">m \geq 3 </tex-math></inline-formula>, there exists a network which has no vector linear solution if the dimension is less than <inline-formula> <tex-math notation="LaTeX">m-1 </tex-math></inline-formula> but has a vector linear solution for all dimensions greater than or equal to <inline-formula> <tex-math notation="LaTeX">m-1 </tex-math></inline-formula>.